The OEIS is supported by the many generous donors to the OEIS Foundation.

 Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”). Other ways to Give
 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A300153 Square array T(n, k) read by antidiagonals upwards, n > 0 and k > 0: T(n, k) is the number of parts inscribed in a rose or rhodonea curve with polar coordinates r = cos(t * (k/n)). 2
 1, 4, 4, 2, 1, 3, 8, 12, 12, 8, 3, 4, 1, 4, 5, 12, 20, 24, 24, 20, 12, 4, 2, 9, 1, 10, 3, 7, 16, 28, 4, 40, 40, 4, 28, 16, 5, 8, 12, 12, 1, 12, 14, 8, 9, 20, 36, 48, 56, 60, 60, 56, 48, 36, 20, 6, 3, 2, 4, 20, 1, 21, 4, 3, 5, 11, 24, 44, 60, 72, 80, 84, 84, 80 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS For any real p > 0, the rose or rhodonea curve with polar coordinates r = cos(t * p): - is dense in the unit disk when p is irrational, - is closed when p is rational, say p = u/v in reduced form; in that case, the number of parts inscribed in the curve is T(v, u), - see also the illustration in Links section. LINKS Rémy Sigrist, Illustration of the first terms Eric Weisstein's World of Mathematics, Rose Wikipedia, Rose (mathematics) FORMULA T(1, k) = A022998(k). T(n, k) = T(n/gcd(n, k), k/gcd(n, k)). Empirically, when gcd(n, k) = 1, we have the following formulas depending on the parity of n and of k: | k is odd | k is even ----------+--------------------------------+-------------------- n is odd | T(n, k) = k * A029578(n+1) | T(n, k) = 2 * k * n n is even | T(n, k) = 2 * k * A029578(n+1) | N/A EXAMPLE Array T(n, k) begins: n\k| 1 2 3 4 5 6 7 8 9 ---+--------------------------------------------- 1| 1 4 3 8 5 12 7 16 9 2| 4 1 12 4 20 3 28 8 36 3| 2 12 1 24 10 4 14 48 3 4| 8 4 24 1 40 12 56 4 72 5| 3 20 9 40 1 60 21 80 27 6| 12 2 4 12 60 1 84 24 12 7| 4 28 12 56 20 84 1 112 36 8| 16 8 48 4 80 24 112 1 144 9| 5 36 2 72 25 12 35 144 1 10| 20 3 60 20 4 9 140 40 180 11| 6 44 18 88 30 132 42 176 54 ... The following diagram shows the curve for T(2, 1) and the corresponding 4 parts: | ######## ######## ##### ####### ##### ### ### ### ### ### ## | ## ### ## ## ## ## ## # Part #2 # ## ## ## ## ## # ### | ### # -#- - - Part #3 - -#######- - Part #1 - - -#- # ### | ### # ## ## ## ## ## # Part #4 # ## ## ## ## ## ### ## | ## ### ### ### ### ### ##### ####### ##### ######## ######## | CROSSREFS Cf. A022998, A029578. Sequence in context: A196766 A153163 A168455 * A182781 A291085 A193556 Adjacent sequences: A300150 A300151 A300152 * A300154 A300155 A300156 KEYWORD nonn,tabl AUTHOR Rémy Sigrist, Feb 26 2018 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 9 15:12 EST 2022. Contains 358700 sequences. (Running on oeis4.)